↑

Properties

Label 2-637-91.4-c1-0-32
Degree $2$
Conductor $637$
Sign $-0.0485 + 0.998i$
Analytic cond. $5.08647$
Root an. cond. $2.25532$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.10i·2-s + (1.13 + 1.95i)3-s − 2.44·4-s + (3.11 − 1.80i)5-s + (4.13 − 2.38i)6-s + 0.948i·8-s + (−1.05 + 1.83i)9-s + (−3.79 − 6.57i)10-s + (−0.767 + 0.443i)11-s + (−2.76 − 4.79i)12-s + (1.17 − 3.40i)13-s + (7.05 + 4.07i)15-s − 2.89·16-s − 4.96·17-s + (3.86 + 2.23i)18-s + (2.06 + 1.18i)19-s + ⋯
L(s)  = 1  − 1.49i·2-s + (0.652 + 1.13i)3-s − 1.22·4-s + (1.39 − 0.805i)5-s + (1.68 − 0.973i)6-s + 0.335i·8-s + (−0.352 + 0.610i)9-s + (−1.20 − 2.08i)10-s + (−0.231 + 0.133i)11-s + (−0.799 − 1.38i)12-s + (0.325 − 0.945i)13-s + (1.82 + 1.05i)15-s − 0.724·16-s − 1.20·17-s + (0.910 + 0.525i)18-s + (0.472 + 0.272i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0485 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0485 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(637\)    =    \(7^{2} \cdot 13\)
Sign: $-0.0485 + 0.998i$
Analytic conductor: \(5.08647\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{637} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 637,\ (\ :1/2),\ -0.0485 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52454 - 1.60047i\)
\(L(\frac12)\) \(\approx\) \(1.52454 - 1.60047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + (-1.17 + 3.40i)T \)
good2 \( 1 + 2.10iT - 2T^{2} \)
3 \( 1 + (-1.13 - 1.95i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.11 + 1.80i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.767 - 0.443i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.96T + 17T^{2} \)
19 \( 1 + (-2.06 - 1.18i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.85T + 23T^{2} \)
29 \( 1 + (0.640 - 1.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.33 - 4.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.63iT - 37T^{2} \)
41 \( 1 + (10.4 + 6.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.82 + 3.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.58 + 1.49i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.46 - 4.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.32iT - 59T^{2} \)
61 \( 1 + (0.769 - 1.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.29 - 4.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.58 - 3.22i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.19 - 3.57i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.378 + 0.656i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.76iT - 83T^{2} \)
89 \( 1 - 3.61iT - 89T^{2} \)
97 \( 1 + (-0.401 + 0.231i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.15720050459962978602628607084, −9.861224270444366178889806085605, −8.901722835163239033620366703860, −8.549229775994661188003218122920, −6.61905533941396755414785293661, −5.22052631777779372735317693101, −4.61703600549027897703553208411, −3.38919079697767821746681118269, −2.55487368123319036028351170912, −1.30332913014067216426155904594, 1.86393603104746667352016878593, 2.75697022590249703767590092703, 4.70600628455449478456888122627, 5.90421048387710051472475827730, 6.60303161652109579199081350267, 6.98458713756351033409033724276, 7.923303152217576145969022807728, 8.853908743998066245921440022960, 9.457236701562841096002797269121, 10.69870913167078478313868132156

Graph of the $Z$-function along the critical line